Am I entitled to demand the sum from (S) that (P) pledged to pay on behalf of (C), and was I wrong in taking the oath in the stated form, and what is the ruling on that?

The one who arbitrates disputes is the Sharia judge or the Sharia arbitrator. It appears that (S) committed to settling your debts as a guarantor, and therefore he is obligated to pay. Guaranty is legally binding according to the Quran, Sunnah, and scholarly consensus (Ijma').

If (S)'s commitment was a promise, then he must fulfill it, especially if the promise caused harm to the promisee. If he does not fulfill it, he should be asked either to implement the promise or to compensate for the damage. There is no harm in you taking an oath and demanding (S) to fulfill his obligation.